polynomial functions - bhsmathfeller

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Lesson 4-1

Polynomial Functions

Explore Power Functions Online Activity Use graphing technology to complete the Explore.

INQUIRY How do the coefficient and degree of a function of the form f(x) = axn affect its end behavior?

Learn Graphing Power Functions

A power function is any function of the form f(x) = axn where a and n are nonzero real numbers. For a power function, a is the leading coefficient and n is the degree, which is the value of the exponent. A power function with positive integer n is called a monomial function.

Key Concept • End Behavior of a Monomial Function

Degree: evenLeading Coefficient: positiveEnd Behavior:

As x → -∞, f(x) → ∞.As x → ∞, f(x) → ∞.

y

xO

y = x414

Domain: all real numbersRange: all real numbers ≥ 0

Degree: oddLeading Coefficient: positiveEnd Behavior:

As x → -∞, f(x) → -∞.As x → ∞, f(x) → ∞.

y

xO

y = x5

Domain: all real numbersRange: all real numbers

Degree: evenLeading Coefficient: negativeEnd Behavior:

As x → -∞, f(x) → -∞.As x → ∞, f(x) → -∞.

y

xO

y = -x2

Domain: all real numbersRange: all real numbers ≤ 0

Degree: oddLeading Coefficient: negativeEnd Behavior:

As x → -∞, f(x) → ∞.As x → ∞, f(x) → -∞.

y

xO

y = -x3

Domain: all real numbersRange: all real numbers

Today’s Vocabulary power function

leading coefficient

degree

monomial function

polynomial in one variable

standard form of a polynomial

degree of a polynomial

polynomial function

quartic function

quintic function

Talk About It!

Is f(x) = √ __

x a power function? a monomial function? Explain your reasoning.

Today’s Goals● Graph and analyze

power functions.

● Graph and analyze polynomial functions.

Lesson 4-1 • Polynomial Functions 213

Sample answer: f(x) = √

__ x is a power function

because it is equivalent

to f(x) = x 1 __ 2 , and 1 __ 2 is a

nonzero constant real number. It is not a monomial function

because 1 __ 2 is not a

positive integer.

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

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Key Concept • Zeros of Even and Odd Degree Functions

Odd-degree functions will always have at least one real zero. Even-degree functions may have any number of real zeros or no real zeros at all.

Example 1 End Behavior and Degree of a Monomial Function

Describe the end behavior of f(x) = −2x3 using the leading coefficient and degree, and state the domain and range.

The leading coefficient of f (x) is , which is negative.

The degree is , which is odd.

Because the leading coefficient is negative and the degree is odd, the end behavior is that as x → -∞, f(x) → and as x → ∞, f(x) → .

Because the leading coefficient is negative and the degree is odd, the domain and range are .

CheckDescribe the end behavior, domain, and range of f(x) = -10x6.

end behavior: As x → -∞, f(x) → and as x → ∞, f(x) → .

domain: range:

Example 2 Graph a Power Function by Using a Table

PRESSURE For water to flow through a garden hose at a certain rate in gallons per minute (gpm), it needs to have a specific pressure in pounds per square inch (psi). Through testing and measurement, a company that produces garden hoses determines that the pressure P given the flow rate F is defined by P(F) = 3 __ 2 F2. Graph the function P(F), and state the domain and range.

Steps 1 and 2 Find a and n. Then state the domain and range.

For P(F) = 3 __ 2 F2, a = , and n = .

Because a is positive and n is even, the domain is and the range is all real numbers .

Steps 3–5 Create a table of values and graph the ordered pairs.

F 3 __ 2 F2 P(F)

-2 3 __ 2 (-2)2

-1 3 __ 2 (-1)2 1.5

0 3 __ 2 (0)2

1 3 __ 2 (1)2

2 3 __ 2 (2)2 6

Go Online You can complete an Extra Example online.

Think About It!

Interpret the domain and range given the context of the situation.

Go OnlineYou can watch a video to see how to graph power functions on a TI-84.

−4−2−6−8

810121416

642

2 4 6 8

y

xO

Pres

sure

(psi

)

Flow Rate (gpm)

Your Notes

214 Module 4 • Polynomials and Polynomial Functions

Sample answer: Because it is unreasonable to have negative flow rate or negative pressure from a garden hose, both the domain and range are constricted to values greater than or equal to zero.

-2

∞-∞

-∞ -∞

3

2

≥ 0all real numbers

all real numbers

3 __ 2

6

0

1.5

all real numbers ≤ 0all real numbers


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Think About It!

Jamison says the leading coefficient of 4x2 - 3 + 2x3 - x is 4. Do you agree or disagree? Justify your reasoning.

Think About It!

If a polynomial function has a leading coefficient of 4, can you determine its end behavior? Explain your reasoning.

Explore Polynomial Functions Online Activity Use graphing technology to complete the Explore.

Learn Graphing Polynomial Functions

A polynomial in one variable is an expression of the form anxn + an - 1x

n - 1 + . . . a2x2 + a1x + a0, where an ≠ 0, an - 1, a1, and a0 are real numbers, and n is a nonnegative integer. Because the terms are written in order from greatest to least degree, this polynomial is written in standard form. The degree of a polynomial is n and the leading coefficient is an.

A polynomial function is a continuous function that can be described by a polynomial equation in one variable. You have learned about constant, linear, quadratic, and cubic functions. A quartic function is a fourth-degree function. A quintic function is a fifth-degree function. The degree tells you the maximum number of times that the graph of a polynomial function intersects the x-axis.

Example 3 Degrees and Leading Coefficients

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why.

a. 2x4 - 3x3 - 4x2 - 5x + 6a degree: leading coefficient:

b. 7x3 - 2 degree: leading coefficient:

c. 4x2 - 2xy + 8y2 This is not a polynomial in one variable. There are two variables, x and y.

d. x5 + 12x4 - 3x3 + 2x2 + 8x + 4 degree: leading coefficient:

CheckSelect the degree and leading coefficient of 11x3 + 5x2 - 7x - 6 __ x .

A. degree: 3, leading coefficient: 11

B. degree: 11, leading coefficient: 3

C. This is not a polynomial in one variable. There are two variables, x and y.

D. This is not a polynomial in one variable. The term 6 __ x has the variable with an exponent less than 0.

Watch Out!

Leading Coefficients If the term with the greatest degree has no coefficient shown, as in part d, the leading coefficient is 1.

Go OnlineYou can learn how to graph a polynomial function by watching the video online. INQUIRY How is the degree of a function

related to the number of times its graph intersects the x-axis?


No; sample answer: I would need to know the degree of the function to determine its end behavior.

Disagree; sample answer: The term with the greatest degree is 2x3, and the coefficient of that term is 2.

4 2

73

5 1

D


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Example 4 Evaluate and Graph a Polynomial Function

SUN The density of the Sun, in grams per centimeter cubed, expressed as a percent of the distance from the core of the Sun to its surface can be modeled by the function f(x) = 519x4 - 1630x3 + 1844x2 + 155, where x represents the percent as a decimal. At the core x = 0, and at the surface x = 1.

Part A Evaluate the function.

Find the core density of the Sun at a radius 60% of the way to the surface.

Because we need to find the core density at a radius 60% of the way to the surface, x = 0.6. So, replace x with 0.6 and simplify.

f(x) = 519x4 - 1630x3 + 1844x2 - 889x + 155

= 519(0.6)4 - 1630(0.6)3 + 1844(0.6)2 - 889(0.6) + 155

= 67.2624 - 352.08 + 663.84 - 533.4 + 155

= g __

cm3

Part B Graph the function.

Sketch a graph of the function.

Substitute values of x to create a table of values. Then plot the points, and connect them with a smooth curve.

x f(x)

0.1 82.9619

0.2

0.3

0.4 3.4064

0.5

0.7 1.7819

0.8

0.9

CheckCARDIOLOGY To help predict heart attacks, doctors can inject a concentration of dye in a vein near the heart to measure the cardiac output in patients. In a normal heart, the change in the concentration of dye can be modeled by f(x) = -0.006x4 + 0.140x3 - 0.053x2 + 1.79x, where x is the time in seconds.

Part A Find the concentration of dye after 5 seconds.

f(5) =

Think About It!

What values of x make sense in the context of the situation? Justify your reasoning.

Study Tip

Axes Labels Notice that the x-axis is measuring the percent of the radius, not the actual length of the radius.


80

60

40

20

0.2 0.4

Percentage of Radius

Den

sity

(g/c

m3 )

0.6 0.8

y

x0


0.6224

38.750414.4539

0.1875

1.9824 0.7859

21.375

Sample answer: Because x is a percent, only values between 0 and 1, inclusively, make sense in the context of the situation.


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Part B Select the graph of the concentration of dye over 10 seconds.

A. B.

C. D.

80

60

40

20

2 4

Seconds

Con

cent

ratio

n of

Dye

6 8

y

x0

16

12

8

4

2 4

Seconds

Con

cent

ratio

n of

Dye

6 8

y

x0

80

60

40

20

2 4

Concentration of Dye

Seco

nds

6 8

y

x0

16

12

8

4

2 4Concentration of Dye

Seco

nds

6 8

y

x0

Example 5 Zeros of a Polynomial Function

Use the graph to state the number of real zeros of the function.

The real zeros occur at x = -2, 1, and 4, so

there are real zeros.

CheckUse the graph to state the number of real zeros of the function.

−4−2−6

160200

1208040

2 4 6

y

xO

The function has real zero(s).

−4−2−6−8

8642

2 4 6 8

−4−6−8

y

xO

Study Tip

Zeros The real zeros occur at values of x where f(x) = 0, or where the polynomial intersects the x-axis. Recall that odd-degree polynomial functions have at least one real zero and even-degree polynomial functions have any number of real zeros. So, the minimum number of times that an odd-degree polynomial intersects the x-axis is 1, and the minimum number of times that an even-degree polynomial intersects the x-axis is 0.



three

4

A


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Think About It!

Find the domain and range of f(x). Does g(x) have the same domain and range? Explain.

Study Tip

Zeros The zeros of a polynomial function are the x-coordinates of the points at which the graph intersects the x-axis.


Example 6 Compare Polynomial Functions

Examine f(x) = x3 + 2x2 - 3x and g(x) shown in the graph.

Part A Graph f(x).

Substitute values for x to create a table of values. Then plot the points, and connect them with a smooth curve.

Part B Analyze the extrema.

Which function has the greater relative maximum?

f(x) has a relative maximum at approximately y = 6, and g(x) has a relative maximum between y = 2 and y = 3. So, has the greater relative maximum.

Part C Analyze the key features.Compare the zeros, x- and y-intercepts, and end behavior of f(x) and g(x).

zeros:

f(x): , ,

g(x): The graph appears to intersect the x-axis at , , ,

intercepts:

f(x): x-intercepts: , , ; y-intercept: 0

g(x): x-intercepts: , , , ; y-intercept: 2

end behavior:

f(x): As x → -∞, f(x) → , and as x → ∞, f(x) → .

g(x): As x → -∞, g(x) → , and as x → ∞, g(x) → .

Pause and Reflect

Did you struggle with anything in this lesson? If so, how did you deal with it?

Record your observations

here.

x f(x)-3 0-2-1 401 023 36

−1−2

4321

1 2

−2−3−4

y

xO

g(x)

−2−1−3−4

8642

1 2 3 4

−4−6−8

y

xO

f(x)


f(x)

-3-1 1-0.5

0 1

2

-3

-∞ ∞

∞ ∞

-1 1-0.50 1

2

The domain and range of f(x) are both all real numbers. No; sample answer: The domain of g(x) is also all real numbers, but the range is all real numbers greater than or equal to the minimum, which is between y = −2 and y = −3.

See students’ observations.

6

0

10


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